Consider the linear system of equations below 3x1 − x2 + x3 = 1 3x1 +...

Consider the linear system of equations below
3x1 − x2 + x3 = 1
3x1 + 6x2 + 2x3 = 0
3x1 + 3x2 + 7x3 = 4
i. Use the Gauss-Jacobi iterative technique with x
(0) = 0 to find
approximate solution to the system above up to the third step
ii. Use the Gauss-Seidel iterative technique with x
(0) = 0 to find
approximate solution to the third step

Expert Solution

Colby Messinger answered 2 years ago

Consider the linear system of equations 2x1 − 6x2 − x3 = −38 −3x1 − x2...

Consider the linear system of equations 2x1 − 6x2 − x3 = −38 −3x1 − x2 + 7x3 = −34 −8x1 + x2 − 2x3 = −20 With an initial guess x (0) = [0, 0, 0]T solve the system using Gauss-Seidel method.

1. Solve the following system: 2x1- 6x2- x3 = -38 -3x1–x2 +7x3 = -34 -8x1 +x2...

1. Solve the following system: 2x1- 6x2- x3 = -38 -3x1–x2 +7x3 = -34 -8x1 +x2 – 2x3 = -20 By: a. LU Factorization b. Gauss-Siedel Method, error less that10-4 Hint (pivoting is needed, switch rows).

Consider the following linear optimization model. Z = 3x1+ 6x2+ 2x3 st 3x1 +4x2 + x3...

Consider the following linear optimization model. Z = 3x1+ 6x2+ 2x3 st 3x1 +4x2 + x3 ≤2 x1+ 3x2+ 2x3 ≤ 1 X1, x2, x3 ≥0 (10) Write the optimization problem in standard form with the consideration of slack variables. (30) Solve the problem using simplex tableau method. (10) State the optimal solution for all variables.

(a) Consider three positive integers, x1, x2, x3, which satisfy the inequality below: x1 +x2 +x3...

(a) Consider three positive integers, x1, x2, x3, which satisfy the inequality below: x1 +x2 +x3 =17. (1) Let’s assume each element in the sample space (consisting of solution vectors (x1, x2, x3) satisfying the above conditions) is equally likely to occur. For example, we have equal chances to have (x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the probability the events x1 +x2 ≤8occurs,i.e.,P(x1 +x2 ≤8|x1 +x2 +x3 =17andx1,x2,x3 ∈Z+)(Z+...

Is the following map linear? a) F(x1,x2,x3)=(0,0) b) L:R2→R2 defined by L(x1,x2)=(3x1−2x2,x2) c) f:R→R defined by...

Is the following map linear? a) F(x1,x2,x3)=(0,0) b) L:R2→R2 defined by L(x1,x2)=(3x1−2x2,x2) c) f:R→R defined by f(x)=2x

Example #2: Write the following set of four linear equations with 4 unknowns x1, x2, x3,...

Example #2: Write the following set of four linear equations with 4 unknowns x1, x2, x3, and x4 in the matrix form. Solve the equations using MATLAB. 0.1 x1+ 2.3 x2 + 3x3 + 4x4 =1 x1+ 3x2 -7x3 +5x4 =2 3x1+2x2+7x3 =3 x1 +2x2 +x3 +10x4=0 (b)Roots of Polynomials: In order to obtain the roots of a polynomial with the coefficients a1,a2,a3 ,... (where a1 is the coefficient of the highest power, and so on in a descending order)...

Solve the following set of equations with LU factorization with pivoting: 3x1 -2x2 + x3 =...

Solve the following set of equations with LU factorization with pivoting: 3x1 -2x2 + x3 = -10 2x1 + 6x2- 4x3 = 44 -8x1 -2x2 + 5x3 = -26 Please show all steps

Given the following LP max z = 2x1 + x2 + x3 s. t. 3x1 -...

Given the following LP max z = 2x1 + x2 + x3 s. t. 3x1 - x2 <= 8 x2 +x3 <= 4 x1,x3 >= 0, x2 urs (unrestricted in sign) A. Reformulate this LP such that 1)All decision variables are non-negative. 2) All functional constraints are equality constraints B. Set up the initial simplex tableau. C. Determine which variable should enter the basis and which variable should leave.

Consider a general system of linear equations with m equations in n variables, called system I....

Consider a general system of linear equations with m equations in n variables, called system I. Let system II be the system obtained from system I by multiplying equation i by a nonzero real number c. Prove that system I and system II are equivalent.

Consider a homogeneous system of linear equations with m equations and n variables. (i) Prove that...

Consider a homogeneous system of linear equations with m equations and n variables. (i) Prove that this system is consistent. (ii) Prove that if m < n then the system has infinitely many solutions. Hint: Use r (the number of pivot columns) of the augmented matrix.

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